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On a Markov chain related to the individual lengths in the recursive construction of Kingman's coalescent

Linglong Yuan

Abstract

Kingman's coalescent is a widely used process to model sample genealogies in population genetics. Recently there have been studies on the inference of quantities related to the genealogy of additional individuals given a known sample. This paper explores the recursive (or sequential) construction which is a natural way of enlarging the sample size by adding individuals one after another to the sample genealogy via individual lineages to construct the Kingman's coalescent. Although the process of successively added lineage lengths is not Markovian, we show that it contains a Markov chain which records the information of the successive largest lineage lengths and we prove a limit theorem for this Markov chain.

On a Markov chain related to the individual lengths in the recursive construction of Kingman's coalescent

Abstract

Kingman's coalescent is a widely used process to model sample genealogies in population genetics. Recently there have been studies on the inference of quantities related to the genealogy of additional individuals given a known sample. This paper explores the recursive (or sequential) construction which is a natural way of enlarging the sample size by adding individuals one after another to the sample genealogy via individual lineages to construct the Kingman's coalescent. Although the process of successively added lineage lengths is not Markovian, we show that it contains a Markov chain which records the information of the successive largest lineage lengths and we prove a limit theorem for this Markov chain.
Paper Structure (12 sections, 15 theorems, 54 equations, 2 figures)

This paper contains 12 sections, 15 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. Main results
  4. Proof of Theorem \ref{['thm:markov']}
  5. Aldous's construction
  6. Identify $(L_n)$ from Aldous's construction
  7. Identify $(R,A)$ from Aldous's construction
  8. A simpler way of identifying $(R,A)$ from Aldous's construction
  9. Proof of Theorem \ref{['thm:markov']}
  10. Proof of Theorem \ref{['cor:limit']}
  11. Preliminaries
  12. Proof of Theorem \ref{['cor:limit']}

Key Result

Theorem 1

The process $(R,A)$ is a Markov chain such that

Figures (2)

  • Figure 1: On the left, the recursive construction is up to individual 3, and on the right is up to individual 4. The bold segments are the external branches. On the left, individual 3 is just added and thus $L_3$ is the external branch length of individual 3. On the right, since individual 4 coalesced with individual 3, the external branch length of individual 3 becomes $L_4$ which is smaller than $L_3$.
  • Figure 2: Aldous's construction of Kingman's coalescent. The vertical axis is the time axis. The stick $i$ is at $U_i$ with height $\tau_i$. The $V_i$'s (the crosses) are the locations of individuals. For the 7 individuals in the figure, the partition at time $t$ is $\{ \{2,4,6\}, \{5\},\{1\},\{3,7\}\}.$

Theorems & Definitions (32)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • ...and 22 more